Problem: The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7?
Answer: $175=5^2\cdot7^1$. To arrange the divisors of 175, we can (conveniently) start with the divisor 7. Since $\gcd(5,7)=1$, the two divisors that are adjacent to 7 must be multiples of 7, which means they have to be $5^1\cdot7=35$ and $5^2\cdot7=175$. Thus, the sum of the two integers adjacent to 7 is $35+175=\boxed{210}$.